Compromise is hard whether it's something as complicated as a salary or as trivial as who does the dishes on Tuesday night. The reason is that we're all pretty certain we're right and Scientific American recommends using that righteousness, aka the self-serving bias, to reach a fair compromise by adding in a third party.

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Everyone assumes they're on the right side of the fence when it comes to compromise and because of that reaching a fair agreement is hard. Where a lot of people once thought the best way to reach a good compromise was to to write an essay from the other side's point of view or have both sides list the holes in their argument, behavioral economist George Lowenstein recommends a completely different approach: use that bias to create a solution. Scientific American explains how this works:

[It's] because a strong bias can blind combatants to the idea that a third party could see it any way but their own.

It's not just that I would like at least eight poker chips, but that I believe the abstract idea of fairness is certain to award me at least these eight chips. And you're equally certain you'll get at least the twelve chips at the bottom end of your fairness scale. So we're both happy to let a fair third party make the call, both blithely confident that the outcome will be the one we want. Self-serving bias makes us both likely to agree to arbitration.

You assume you're right and because of that you assume a third party will also see it the same way. However, when a third party comes in they're more likely to split the difference fairly in a way just two people wouldn't.